Metric Space
Contents
1.1. Metric Space¶
Definition 1.1 (Metric Space)
Let be \(X\) a set nonempty. A function \(\rho : X \times X \to \mathbb{R}\) is a distance function if and only if \(\forall x, y, z \in X\) such that:
\(f(x, y) \geq 0\)
\(f(x, y) = 0 \Leftrightarrow x = y\)
\(f(x, y) = f(y, x)\)
\(f(x, y) \leq f(x, z) + f(z, y)\)
In that case a \((X, \rho)\) we called metric space.
Example 1.1
\(X = \mathbb{R}\)
\(x = (x_1, x_2)\), \(y = (y_1, y_2)\)
\(d(x, y) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2}\)
\((\mathbb{R}, d)\) is a m.e.
Example 1.2
\(X = \mathbb{R}\)
\(x = (x_1, x_2)\), \(y = (y_1, y_2)\)
\(d_{\infty}(x, y) = \max \{|x_1 - y_1|, |x_2 - y_2| \}\)
\((\mathbb{R}, d_{\infty})\) is a m.e.
Example 1.3
\(X = \{ f:[0,1] \to \mathbb{R}\ | \ f \text{ is continuous}\}\)
\(x = (x_1, x_2)\), \(y = (y_1, y_2)\)
\(\rho(f, g) = \int_0^1 |f(x) - g(x)| dx\)
\(\sigma(f, g) = \max \{ |f(x) - g(x)|, \ x \in [0, 1] \}\)
1.1.1. Varous definitions¶
Definition 1.2 (Neighbourhood)
Let be \(X\) a metric space and \(p \in X\). A neighborhood of \(p\) is a set \(N_r(p)\) consisting of all \(q\) such that \(d(p, q) < r\), for some \(r > 0\). The number \(r\) is called the radius of \(N_r(p)\).
Definition 1.3 (Limit Point)
Let be \(X\) a metric space and \(p \in X\). A point \(p\) is a limit point of the set \(E \subset X\) if every neighborhood of \(p\) constain a point \(q \neq p\) such that \(q \in E\).
Definition 1.4 (Isolated Point)
Let be \(X\) a metric space, \(p \in X\) and \(E \subset X\). If \(p \in E\) and \(p\) is not limit point of \(E\), then \(p\) is called an isolated point.
Definition 1.5 (Closed)
Let be \(X\) a metric space and \(E \subset X\). \(E\) is closed if every limit point of \(E\) is a point of \(E\).
Definition 1.6 (Interior (point))
Let be \(X\) a metric space, \(p \in X\) and \(E \subset X\). A point \(p\) is an interior point of \(E\) if there is a neighborhood \(N\) of \(p\) such that \(N \subset E\).
Definition 1.7 (Open)
Let be \(X\) a metric space and \(E \subset X\). \(E\) is open if every point of \(E\) is an interior point of \(E\).
Definition 1.8 (Bounded)
Let be \(X\) a metric space, \(p, q \in X\) and \(E \subset X\). \(E\) is bounded if there is a real number \(M\) and a point \(p \in X\) such that \(d(p, q) < M\) for all \(p \in E\).
Definition 1.9 (Dense)
Let be \(X\) a metric space and \(E \subset X\). \(E\) is dense in \(X\) if every point of \(X\) is limit point of \(E\), or a point of \(E\) (or both).
Theorem 1.1
Every neighborhood is an open set.
Proof
Consider a neighborhood \(E=N_{p}(p)\), and let \(q\) be any point of \(E\). Then there is a positive real number \(h\) such that
For all points \(s\) such that \(d(q, s)<h\), we have then
so that \(s \in E\). Thus \(q\) is an interior point of \(E\).